minimal prime ideals in noetherian local rings
I can't figure it out. The question is this:
Are there only a finite number of minimal prime ideals in a local noetherian ring?
This question arises in the dimension theory of such rings. An affirmative answer seems to be assumed in the proof I've seen of this proposition:
Let A be such a ring and assume that dim A = d. Let m be the maximal ideal of A. Then there is an m-primary ideal in A generated by d elements.
This is one of the steps towards the dimension theorem asserting the equivalence of three definitions of dimension for local noetherian rings.